Rasmus Hojlund celebrates scoring the winner for Denmark against Portugal. Photo: Reuters
The hosts created plenty of chances and playmaker Christian Eriksen had a first-half penalty saved by Portugal goalkeeper Diogo Costa, who came to the rescue on several occasions, while Eriksen also had a goal-bound shot cleared by Diogo Dalot.
There was little Costa could do about Denmark's goal though as Eriksen picked out Andreas Skov Olsen on the right of the box and he squared the ball for Hojlund to steer it home with his right foot in the 78th minute.
Denmark backed off after scoring and invited Portugal to attack but the home side managed to keep Cristiano Ronaldo's side quiet and take a lead into Sunday's second leg in Lisbon.
The winners of that tie will face either German or Italy in the semi-finals.
The Germans have the advantage after they came from behind to defeat Italy 2-1 at the San Siro in the first leg of their quarter-final, with Leon Goretzka heading the winning goal 14 minutes from time.
The hosts were ahead in the ninth minute when the ball fell to Sandro Tonali who blasted his shot into the corner of the net and while Germany enjoyed more possession in the opening half, Italy looked the more dangerous side.
Germany levelled through substitute Tim Kleindienst's header four minutes after the break and Goretzka's glancing header from a corner earned the visitors the away win.
Meanwhile, Ivan Perisic scored one goal and made another as Croatia beat France 2-0 to seize the upper hand in the first leg of their quarter-final tie.
Perisic crossed for Ante Budimir to head in the opener at the Stadion Poljud in Split after the home side had squandered an early penalty.
With France out of sorts, Perisic made it 2-0 before the break and a Croatia side captained by Luka Modric saw out the victory to take to Paris for the return match on Sunday.
It was not a good night for France as captain Kylian Mbappe made his comeback having not featured in either of the last two international windows, in October and November.
Mbappe did have chances, notably seeing a deflected second-half shot from Ousmane Dembele’s cutback kept out by a fine Dominik Livakovic save.
The Real Madrid superstar has now failed to find the net in his last six appearances for his country, since scoring a penalty in a 1-1 draw with Poland at Euro 2024.
On the other side of the draw, Spain snatched a stoppage-time equaliser against 10-man Netherlands to secure a 2-2 draw on Thursday that kept up a two-year unbeaten streak and gave the holders the advantage in their quarter-final tie.
Substitute Mikel Merino, a makeshift striker at club level for Arsenal, proved his ability in front of goal for his country too with a close-range finish three minutes into added time.
The Dutch had fought back after going behind in the ninth minute with goals from Cody Gakpo and Tijjani Reijnders but finished with 10 men after teenage defender Jorrel Hato was sent off eight minutes from time for a studs-up challenge.
Spain, who had not lost in their previous 21 competitive internationals, opened the scoring through Nico Williams.